rep
yr     1   2   7
  4  104 110  92
  5  104 110  92
  7  104 110  92
  8  102 110  91
  10 104 110  93
  11 103 110  92
  13 100 110  91
  14 105 110  93
  15 105 110  93

• There is a strong temporal trend that needs to be accounted for

• Family of orthogonal polynomials dense in \(\mathscr{L}_2\)
• Any regular curve can be approximated as much as needed by taking a linear combination of polynomials up to a high-enough order

\[ \begin{aligned} y_{ij} = & \mathrm{X}_i \beta + \Sigma_{k = 0}^{\mathrm{ord}} a_{ik} \mathcal{L}_k(j) + \varepsilon_{ij} \\ [a_0', \ldots, a_{\mathrm{ord}}']' \sim & N(0, \Sigma_{\mathrm{ord}} \otimes \mathrm{A})\\ \varepsilon \sim & N(0, \sigma_e^2) \end{aligned} \]

• The Breeding Value of an individual is a function of an environmental variable (temp., precip., …)

• This function is parameterised as a linear combination of Legendre orthogonal polynomials of order up to a given order

• Each individual is described by order + 1 coefficients

• Functional Breeding Values for each individual